Learning Outcomes
- Explain the meaning of a higher-order derivative
The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of y=f(x)y=f(x) can be expressed in any of the following forms:
It is interesting to note that the notation for d2ydx2 may be viewed as an attempt to express ddx(dydx) more compactly. Analogously,
ddx(ddx(dydx))=ddx(d2ydx2)=d3ydx3
Example: Finding a Second Derivative
For f(x)=2x2−3x+1, find f″(x).
Try It
Find f″(x) for f(x)=x2.
Watch the following video to see the worked solution to the above Try It.
Example: Finding Acceleration
The position of a particle along a coordinate axis at time t (in seconds) is given by s(t)=3t2−4t+1 (in meters). Find the function that describes its acceleration at time t.
Watch the following video to see the worked solution to Example: Finding Acceleration.
Try It
For s(t)=t3, find a(t).
Candela Citations
- 3.2 The Derivative as a Function. Authored by: Ryan Melton. License: CC BY: Attribution
- Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/details/books/calculus-volume-1. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction